This document is the Reference Manual of version 8.3pl3 of the Coq proof assistant. A companion volume, the Coq Tutorial, is provided for the beginners. It is advised to read the Tutorial first. A book [14] on practical uses of the Coq system was published in 2004 and is a good support for both the beginner and the advanced user.

The Coq system is designed to develop mathematical proofs, and especially to write formal specifications, programs and to verify that programs are correct with respect to their specification. It provides a specification language named Gallina. Terms of Gallina can represent programs as well as properties of these programs and proofs of these properties. Using the so-called Curry-Howard isomorphism, programs, properties and proofs are formalized in the same language called Calculus of Inductive Constructions, that is a λ-calculus with a rich type system. All logical judgments in Coq are typing judgments. The very heart of the Coq system is the type-checking algorithm that checks the correctness of proofs, in other words that checks that a program complies to its specification. Coq also provides an interactive proof assistant to build proofs using specific programs called tactics.

All services of the Coq proof assistant are accessible by interpretation of a command language called the vernacular.

Coq has an interactive mode in which commands are interpreted as the user types them in from the keyboard and a compiled mode where commands are processed from a file.

These two modes are documented in Chapter 13.

Other modes of interaction with Coq are possible: through an emacs shell window, an emacs generic user-interface for proof assistant (ProofGeneral [1]) or through a customized interface (PCoq [129]). These facilities are not documented here. There is also a Coq Integrated Development Environment described in Chapter 15.

How to read this book

This is a Reference Manual, not a User Manual, then it is not made for a continuous reading. However, it has some structure that is explained below.

At the end of the document, after the global index, the user can find specific indexes for tactics, vernacular commands, and error messages.

List of additional documentation

This manual does not contain all the documentation the user may need about Coq. Various informations can be found in the following documents:

A companion volume to this reference manual, the Coq Tutorial, is aimed at gently introducing new users to developing proofs in Coq without assuming prior knowledge of type theory. In a second step, the user can read also the tutorial on recursive types (document
The fifth part (the Addendum) of the Reference Manual is distributed as a separate document. It contains more detailed documentation and examples about some specific aspects of the system that may interest only certain users. It shares the indexes, the page numbers and the bibliography with the Reference Manual. If you see in one of the indexes a page number that is outside the Reference Manual, it refers to the Addendum.
A text file INSTALL that comes with the sources explains how to install Coq.
The Coq standard library
A commented version of sources of the Coq standard library (including only the specifications, the proofs are removed) is given in the additional document